Quantum Lower Bounds by Polynomials

We examine the number T of queries that a quantum network requires to compute several Boolean functions on f0;1gN in the black-box model. We show that, in the blackbox model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the socalled polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized an...

• Type: Conference
• Year: 1998
• Where: FOCS
• Authors: Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, Ronald de Wolf
• Keywords: Boolean Function, FOCS 1998, Quantum, Quantum Extension, Theoretical Computer Science